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Ukrainian Mathematical Journal

, Volume 66, Issue 5, pp 645–665 | Cite as

On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

  • F. G. Abdullayev
  • P. Özkartepe
Article
Let G ⊂  be a finite region bounded by a Jordan curve L := ∂G, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \)), let Δ := {w : |w| > 1}, and let w = Φ(z) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let h(z) be a weight function and let A p (h,G), p > 0 denote a class of functions f analytic in G and satisfying the condition
$$ {\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }} $$
where σ is a two-dimensional Lebesgue measure.
Moreover, let P n (z) be an arbitrary algebraic polynomial of degree at most n ∈ ℕ. The well-known Bernstein–Walsh lemma states that
$$ \begin{array}{cc}\hfill \left|{P}_n(z)\right|\le {\left|\varPhi (z)\right|}^n{\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)},\hfill & \hfill z\in \Omega .\hfill \end{array} $$
(*)

In this present work we continue the investigation of estimation (*) in which the norm \( {\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)} \) is replaced by \( {\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0 \), for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.

Keywords

Orthogonal Polynomial Quasiconformal Mapping Jordan Curve Algebraic Polynomial Unbounded Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • F. G. Abdullayev
    • 1
  • P. Özkartepe
    • 1
  1. 1.Mersin UniversityMersinTurkey

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